PanelGP makes its strongest trading case through a better covariance matrix. Return forecasts offer less than the headline Sharpe implies, a weakness Ruglioni and Filipovic partly acknowledge. They write that IPCA performs at least as well on squared error when models are re-estimated annually. Their defence rests on ranking stocks. Even there, annual re-estimation leaves a thin advantage: rank correlation reaches 4.84% for PanelGP versus 3.69% for rolling IPCA with an intercept. The statistic of 2.29 fails the Holm adjustment. The abstract's claim of significantly higher Sharpe ratios than IPCA holds under fixed training, where the statistics are 3.83 against IPCA and 3.14 against its intercept version.
A shared map for returns and risk
A conditional factor model uses the same loadings to forecast returns and covariance. Expected returns multiply those loadings by expected factors; covariance between stocks combines their loadings with the factor covariance. Making loadings a function of firm characteristics lets one estimated map serve both forecasts.
PanelGP assigns Gaussian-process priors to two maps. One turns 137 firm characteristics into five loadings. Each month, the characteristics are ranked into [-0.5, 0.5], with missing values set to zero. The other map forecasts next month's expected factors from six macro variables, called the context in the paper. These are market return, term spread, default spread, log realized market variance, average correlation and lagged inflation.
An asset-specific shock to realized loadings contributes only idiosyncratic variance. A shock to realized factors is shared across stocks at each date and generates covariance. With affine loadings and a constant factor kernel, the model has IPCA's structure, apart from the loading innovation and GP priors. IPCA is Kelly et al.'s affine map from characteristics to loadings. RBF kernels instead allow both maps to be smooth and nonlinear.
The authors estimate the model through sparse variational inference, using 160 inducing points per component. They retain a joint Gaussian posterior over loadings and training factors. A predicted return is a quadratic form in that Gaussian state, which gives the predictive covariance a closed form. Uncertainty about both estimated maps therefore enters portfolio risk directly. Fitting five factors takes about twelve minutes on one H100.
Historical S&P 500 membership from CRSP defines the test universe: 173,740 stock-months over 1995 to 2024, with 465 to 497 stocks each month. The authors form three portfolios monthly. They scale the tangency portfolio to unit gross exposure and allow shorts. They also construct a global minimum-variance (GMV) portfolio and an equal-weight top-minus-bottom decile long-short.
What survives annual re-estimation?
The evidence for nonlinear loadings is persuasive. Switching from affine to RBF loadings improves rank correlation and log score with either factor kernel. Under a constant factor kernel, the Newey-West statistics are -3.39 and -3.31; under an RBF kernel, they are -3.37 and -3.70. All survive the Holm adjustment.
Allowing the factor forecast to depend on macro variables raises predictive R2, rank, log score and both Sharpe ratios. The tests cannot distinguish those increases from zero. With affine loadings, the tangency statistic is -2.01. With RBF loadings, however, GMV volatility falls significantly from 14.7% to 14.3%, with a statistic of -4.36.
For the fixed fit over 1960 to 1989, RBF loadings and RBF factors deliver a tangency Sharpe of 1.55. IPCA returns 0.84, and IPCA with an intercept returns 1.17, all before costs. Thirty-five years with frozen parameters makes for a weak benchmark.
Annual re-estimation over a 360-month window is the fairer comparison. PanelGP then earns a tangency Sharpe of 1.90, against 1.74 for rolling IPCA with an intercept. The test statistic is 1.51.
The gap is insignificant.
The authors acknowledge this directly: "the tests cannot separate these Sharpe ratios, but they do separate the log scores and the minimum-variance volatilities." The evidence supports their "but". PanelGP's log score is 1.130 versus 1.083, with a statistic of 5.82. GMV volatility is 12.3% versus 13.2%, with a statistic of 2.73. Both differences are significant.
Other comparisons support the covariance interpretation. An intercept alone takes rolling IPCA's tangency Sharpe from 0.99 to 1.74. In the fixed-training results, using RBF loadings and RBF factors raises tangency Sharpe from 1.31 at three factors to 1.55 at five. Rank correlation, meanwhile, levels off after two or three. The additional factors improve the covariance matrix, bringing the portfolio gain with them.
The predictive R2 results are harder to read. Under rolling estimation, pooled predictive R2 falls to 0.203%, below rolling IPCA's 0.644% and its intercept version's 0.588%. The decline from 0.753% is insignificant, with a statistic of 1.34. It comes from the level of the forecasts. The monthly average forecast contributes 0.744 pp with fixed training and 0.039 pp with rolling estimation. Contributions from deviations around that average increase from 0.009 to 0.164 pp.
Compressed stock variances
With annual re-estimation, predicted variances rank risk across months at 58.3% Spearman and across stocks within months at 26.9%. Their levels are compressed.
The monthly measure correlates the loading-innovation term with mean squared forecast error each month, across 360 months. For stocks, the measure is the within-month rank correlation of the factor-innovation term, averaged across 1,800 month-seed pairs. Realized errors fan out faster than predicted risk. The model assigns too little variance to the riskiest quintiles and too much to the safest group of stocks.
Part of the cause is structural. At each date, three idiosyncratic terms are the same for every stock: the loading-innovation term, the product of the two innovation variances, and observation noise. This common floor cannot follow stock-specific idiosyncratic risk. The authors acknowledge the compression in the abstract. They also suggest that the variational approximation's reverse KL divergence may explain part of it. Their proposed remedy makes loading-innovation variance or noise depend on characteristics, preserving the closed form. The paper does not implement that change.
Sizing positions from the absolute level of a single stock's predicted volatility carries this compression into the book. Separately, the predictive density's non-Gaussian shape adds about 0.022 to log score per observation, largely through errors beyond three standard deviations. Its tails decay at least exponentially by construction. Power-law tails are therefore outside what the density can produce.
Costs, and a weaker 2010-2024
Charging 10 basis points per dollar traded leaves rolling PanelGP's tangency portfolio with a 1.52 Sharpe. Monthly turnover is 1.06, and break-even cost is 49.7 bp. The long-short earns 0.91 after costs on turnover of 1.38, breaking even at 57.3 bp. Rolling IPCA with an intercept earns 1.35 and 0.72 after costs. The authors acknowledge that flat proportional costs leave out spreads, shorting fees and price impact. A book trading more than its gross exposure each month needs that qualification.
Performance weakens in the later sample. Rolling tangency Sharpe drops from 2.62 over 1995-2009 to 1.16 over 2010-2024. Long-short Sharpe drops from 1.36 to 0.78, while rank correlation falls from 6.16% to 3.53%. Both PanelGP and IPCA share the decline. Rolling PanelGP still has the highest tangency Sharpe in each half: over 2010-2024, it earns 1.16 against 1.04 for rolling IPCA with an intercept.
The S&P 500 earns 0.34 and 0.93 over the same halves. Comparing the later index Sharpe of 0.93 with PanelGP's 1.16 gives a gap of 0.23. This is less than one single-Sharpe standard error of 0.26 or more, and the index is not even market-neutral. The paper benchmarks no other covariance estimator. Its IPCA comparisons cover versions with and without an intercept, under both fixed and rolling estimation; the S&P 500 serves only as a reference level.
We could not rerun the study. It requires S&P 500 membership and characteristics going back to 1960. Our price and fundamental coverage does not extend through the 1995 to 2009 half. Substituting a yearly top-500 market-cap screen would change the universe the paper tests.
We would take PanelGP forward as a risk model to compare with an existing covariance stack, while holding off on its return signals. The 2010-2024 comparison is 1.16 vs 1.04, a gap the authors describe as imprecisely estimated. A test of that gap clearing Newey-West would change our mind.